Such a process in known e.g. from WO 92/10270, WO 94/29227 and WO 96/30110. In these prior art processes oxygen is used as an electron acceptor. The oxygen is fed at a limited rate so as to direct the oxidation of sulphide to elemental sulphur rather than to sulphate. The use of oxygen as an electron acceptor, however, requires the presence of an aeration system including a compressor, pipings, spargers, and in most cases a separate reactor. The investment costs for such a system are relatively high, especially when small volumes of water or gas are to be treated or when e.g. high-pressure natural gas is to be desulphurised.